of Physics, Pohang University of Science and Technology (POSTECH), Pohang, 790-784, South Korea 2 [email protected] ∗

[email protected]

http://qopt.postech.ac.kr

Abstract: When two entangled qubits, each owned by Alice and Bob, undergo separate decoherence, the amount of entanglement is reduced, and often, weak decoherence causes complete loss of entanglement, known as entanglement sudden death. Here we show that it is possible to apply quantum measurement reversal on a single-qubit to avoid entanglement sudden death, rather than on both qubits. Our scheme has important applications in quantum information processing protocols based on distributed or stored entangled qubits as they are subject to decoherence. © 2014 Optical Society of America OCIS codes: (270.5585) Quantum information and processing; (270.5565) Quantum communications.

References and links 1. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University, Cambridge, 2000). 2. C. H. Bennett and D. P. DiVincenzo, “Quantum information and computation,” Nature (London) 404, 247–255 (2000). 3. C. H. Bennett, G. Brassard, C. Cr´epeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70, 1895–1899 (1993). 4. D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum teleportation,” Nature (London) 390, 575–579 (1997). 5. N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002). 6. T. Yu and J. H. Eberly, “Finite-time disentanglement via spontaneous emission,” Phys. Rev. Lett. 93, 140404 (2004). 7. M. P. Almeida, F. de Melo, M. Hor-Meyll, A. Salles, S. P. Walborn, P. H. Souto Ribeiro, and L. Davidovich, “Environment-induced sudden death of entanglement,” Science 316, 579–582 (2007). 8. D. A. Lidar, I. L. Chuang, and K. B. Whaley, “Decoherence-free subspaces for quantum computation,” Phys. Rev. Lett. 81, 2594–2597 (1998). 9. P. G. Kwiat, A. J. Berglund, J. B. Altepeter, and A. G. White, “Experimental verification of decoherence-free subspaces,” Science 290, 498–501 (2000). 10. P. Facchi, D. A. Lidar, and S. Pascazio, “Unification of dynamical decoupling and the quantum Zeno effect,” Phys. Rev. A 69, 032314 (2004). 11. S. Maniscalco, F. Francica, R. L. Zaffino, N. L. Gullo, and F. Plastina, “Protecting entanglement via the quantum Zeno effect,” Phys. Rev. Lett. 100, 090503 (2008). 12. J. G. Oliveira, Jr., R. Rossi, Jr., and M. C. Nemes, “Protecting, enhancing, and reviving entanglement,” Phys. Rev. A 78, 044301 (2008).

#214791 - $15.00 USD Received 27 Jun 2014; accepted 11 Jul 2014; published 29 Jul 2014 (C) 2014 OSA 11 August 2014 | Vol. 22, No. 16 | DOI:10.1364/OE.22.019055 | OPTICS EXPRESS 19055

13. J. A. Schreier, A. A. Houck, Jens Koch, D. I. Schuster, B. R. Johnson, J. M. Chow, J. M. Gambetta, J. Majer, L. Frunzio, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, “Suppressing charge noise decoherence in superconducting charge qubits,” Phys. Rev. B 77, 180502(R) (2008). 14. L. Viola, E. M. Fortunato, M. A. Pravia, E. Knill, R. Laflamme, and D. G. Cory, “Experimental realization of noiseless subsystems for quantum information processing,” Science 293, 2059–2063 (2001). 15. D. Kielpinski, V. Meyer, M. A. Rowe, C. A. Sackett, W. M. Itano, C. Monroe, and D. J. Wineland, “A decoherence-free quantum memory using trapped ions,” Science 291, 1013–1015 (2001). 16. M. Koashi and M. Ueda, “Reversing measurement and probabilistic quantum Error correction,” Phys. Rev. Lett. 82, 2598–2601 (1999). 17. A. N. Korotkov and A. N. Jordan, “Undoing weak quantum measurement of a solid-state qubit,” Phys. Rev. Lett. 97, 166805 (2006). 18. Q. Sun, M. Al-Amri, and M. S. Zubairy, “Reversing the weak measurement of an arbitrary field with finite photon number,” Phys. Rev. A 80, 033838 (2009). 19. N. Katz, M. Neeley, M. Ansmann, R. C. Bialczak, M. Hofheinz, E. Lucero, A. O’Connell, H. Wang, A. N. Cleland, J. M. Martinis, and A. N. Korotkov, “Reversal of the weak measurement of a quantum state in a superconducting phase qubit,” Phys. Rev. Lett. 101, 200401 (2008). 20. Y.-S. Kim, Y.-W. Cho, Y.-S. Ra, and Y.-H. Kim, “Reversing the weak quantum measurement for a photonic qubit,” Opt. Express 17, 11978–11985 (2009). 21. A. N. Korotkov and K. Keane, “Decoherence suppression by quantum measurement reversal,” Phys. Rev. A 81, 040103(R) (2010). 22. J.-C. Lee, Y.-C. Jeong, Y.-S. Kim, and Y.-H. Kim, “ Experimental demonstration of decoherence suppression via quantum measurement reversal,” Opt. Express 19, 16309–16316 (2011). 23. Q. Sun, M. Al-Amri, L. Davidovich, and M. S. Zubairy, “Reversing entanglement change by a weak measurement,” Phys. Rev. A 82, 052323 (2010). 24. Y.-S. Kim, J.-C. Lee, O. Kwon, and Y.-H. Kim, “ Protecting entanglement from decoherence using weak measurement and quantum measurement reversal,” Nature Phys. 8, 117–120 (2012). 25. C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schumacher, “Concentrating partial entanglement by local operations,” Phys. Rev. A 53, 2046–2052 (1996). 26. C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, “Purification of noisy entanglement and faithful teleportation via noisy channels,” Phys. Rev. Lett. 76, 722–725 (1996). 27. P. G. Kwiat, S. Barraza-Lopez, A. Stefanov, and N. Gisin,“Experimental entanglement distillation and ‘hidden’ non-locality,” Nature (London) 409, 1014–1017 (2001). ˘ Brukner, and A. Zeilinger, “Entanglement purification for quantum communication,” 28. J.-W. Pan, C. Simon, C. Nature (London) 410, 1067–1070 (2001). 29. J. P. Groen, D. Rist`e, L. Tornberg, J. Cramer, P. C. de Groot, T. Picot, G. Johansson, and L. DiCarlo, “Partialmeasurement backaction and nonclassical weak values in a superconducting circuit,” Phys. Rev. Lett. 111, 090506 (2013). 30. W. K. Wootters,“Entanglement of formation of an arbitrary state of two qubits,” Phys. Rev. Lett. 80, 2245–2248 (1998). 31. Y. H. Shih and C. O. Alley, “New type of Einstein-Podolsky-Rosen-Bohm experiment using pairs of light quanta produced by optical parametric down conversion,” Phys. Rev. Lett. 61, 2921–2924 (1988).

1.

Introduction

Quantum entanglement is an essential resource for quantum information processing including quantum computation [1, 2] and quantum communication [3–5]. However, decoherence due to irreversible interactions with the environment causes degradation of entanglement, and, in some cases, leads to entanglement sudden death (ESD) in which entanglement is completely lost before its subsystem is fully decohered [6, 7]. Since overcoming decoherence is required for practical realization of quantum information protocols, various schemes for suppressing decoherence have been reported [8–15]. Recently, it has been shown that quantum measurement reversal, a set of weak and reversing measurements [16–20], can effectively suppress decoherence [21–24]. Without quantum measurement reversal, decoherence causes the qubit to end-up in a mixed state due to the loss of quantum coherence, while if quantum measurement reversal is introduced as shown in Fig. 1(a), the initial state may be preserved even in the presence of decoherence [21,22]. This decoherence suppression scheme can also be applied to multi-qubit systems, in particular, it has been shown that entanglement of two-qubit states that undergo separate decoherence can be protected by #214791 - $15.00 USD Received 27 Jun 2014; accepted 11 Jul 2014; published 29 Jul 2014 (C) 2014 OSA 11 August 2014 | Vol. 22, No. 16 | DOI:10.1364/OE.22.019055 | OPTICS EXPRESS 19056

Sys.

Env.

Sys.

Env.

Alice Bob

Alice or

?

Bob Fig. 1. (a) Decoherence suppression scheme using quantum measurement reversal for a single qubit state. To suppress decoherence, a set of weak and reversing measurements are performed before and after decoherence, respectively. Note that the size of the spheres corresponds to the population fraction of each level, |0i and |1i. (b) Entanglement of two-qubit states which undergo separate decoherence can be recovered if both Alice and Bob carry out quantum measurement reversal. (c) The situation we considered here: Only Alice performs single-qubit quantum measurement reversal on her subsystem. W : weak measurement, D: decoherence, R: reversing measurement, Sys: system, Env: environment.

using two sets of single-qubit quantum measurement reversal steps, see Fig. 1(b) [24]. In some cases, decoherence affecting on both subsystems of two-qubit entangled states causes complete loss of entanglement, i.e., ESD [6]. Interestingly, even in this severe circumstance, the initial entanglement can be protected by performing quantum measurement reversal on both qubits. Avoiding ESD is essential in entanglement distribution between two distant parties, since once distributed entangled qubits completely lose entanglement, entanglement cannot be distilled between them [25–28]. A question then naturally arises whether preventing an entangled state from completely losing entanglement against ESD is possible by quantum measurement reversal performed only on one of the qubits rather than on both qubits as shown in Fig. 1(c). For instance, between Alice and Bob who initially share two entangled qubits, only Alice performs quantum measurement reversal on her qubit. In this work, we demonstrate that Alice alone can avoid ESD by performing quantum measurement reversal only on her qubit. 2.

Theory

Consider the situation in which a two-qubit entangled state is affected by separate decoherence. Initially, Alice and Bob share two-qubit pure entangled state |Φi = α |00i+ β |11i where |α |2 + |β |2 = 1 as shown in Fig. 2(a). We assume that each qubit independently suffers from amplitude damping decoherence. In general, amplitude damping decoherence arises due to the interaction between the system (S) and the environment (E) and amplitude damping for a single √ qubit can ¯ S |0iE + be represented by the following operation, |0iS |0iE → |0iS |0iE and |1iS |0iE → D|1i #214791 - $15.00 USD Received 27 Jun 2014; accepted 11 Jul 2014; published 29 Jul 2014 (C) 2014 OSA 11 August 2014 | Vol. 22, No. 16 | DOI:10.1364/OE.22.019055 | OPTICS EXPRESS 19057

Alice

Bob

Alice

Bob

Alice

Bob

Alice

Bob

Alice

Bob

Alice

Bob

Fig. 2. (a) Alice and Bob initially share a pure entangled state |Φi. When both subsystems suffer from decoherence, the final state ρD loses its entanglement partially or completely. (b) To suppress decoherence, Alice performs a set of weak measurement WA (p) and reversing measurement RA (pr ) before and after her subsystem undergoes decoherence, respectively. Then, the resulting state ρA becomes closer to the initial state |Φi, i.e. ρA is more entangled than ρD . Note that Bob is not involved in this decoherence suppression scheme.

√ D|0iS |1iE , where 0 ≤ D ≤ 1 is the magnitude of the decoherence and D¯ ≡ 1 − D. Amplitude damping usually describes the energy dissipation process, for instance, it can be observed from the state of an atom which spontaneously decays, the state of a photon in an leaky cavity or an interferometer, and the zero-temperature energy relaxation of a superconducting qubit [19, 29]. Although Alice and Bob initially share a pure entangled state |Φi, the resulting state ρD is not a pure state since each particle suffers from amplitude damping decoherence with strengths of DA and DB , respectively [7, 24]. The concurrence [30] of the final state ρD , CD is given by n p o √ CD = max 0, ΛD ≡ 2 D¯ A D¯ B |β | |α | − DA DB |β | , (1)

where CD = ΛD if ΛD > 0 and CD is always less than the concurrence of the initial state, 2 |αβ |. From Eq. (1), it is clear that for |α | ≥ |β |, CD = 0 only when DA = 1 or DB = 1, meaning that the entanglement exists except the case that the final state ρD is fully damped by decoherence (Di = 1). On the other hand, for |α | < |β |, entanglement completely disappears √ when DA DB > |α /β |, which is clearly less than the maximum level of the decoherence. Thus, ESD occurs only if |α | < |β | [7]. Let us now describe our strategy to avoid ESD as shown in Fig. 2(b). At first, Alice and Bob initially share a pure entangled state |Φi. Then, Alice performs a weak measurement WA (p) on her qubit before her qubit undergoes decoherence, see Fig. 2(b). Here, weak measurement partially collapses a single qubit √ state towards |0iS . The weak measurement operator W (p) can be written as W (p) = |0i h0| + 1 − p|1i h1| where p is the strength of the weak measurement [20]. Since only |1iS state undergoes decoherence, the system states can partially avoid decoherence by reducing the portion of |1iS state with weak measurements. Note that the initial state |Φi is transformed into |Φ′ i by a weak measurement and |Φ′ i is still a pure state. Then each qubit undergoes separate decoherence and |Φ′ i evolves to a mixed state ρ ′ .

#214791 - $15.00 USD Received 27 Jun 2014; accepted 11 Jul 2014; published 29 Jul 2014 (C) 2014 OSA 11 August 2014 | Vol. 22, No. 16 | DOI:10.1364/OE.22.019055 | OPTICS EXPRESS 19058

Alice now performs a reversing measurement RA (pr ) on her qubit as shown in Fig. 2(b). A reversing measurement is also a weak measurement but it partially collapses the state towards the √ opposite direction, i.e., |1iS . The reversing measurement operator is given as R (pr ) = 1 − pr |0i h0| + |1i h1| where pr is the strength of the reversing measurement. The reversing strength we used is pr = p + DA p¯ where p¯ ≡ 1 − p [21]. Here, Bob does not play any role in the whole procedure. Note that the scheme also works well if only Bob carries out single-qubit quantum measurement reversal on his qubit, i.e., WB (p) and RB (pr ). Note that the strength of the reversing measurement which Bob performs on his qubit is pr = p + DB p. ¯ After Alice alone performs the single-qubit quantum measurement reversal on her qubit, the concurrence of the two-qubit state ρA , CA , can be evaluated by ( p ) √ 2 D¯ B |β | |α | − DA DB p¯ |β | CA = max 0, ΛA = , (2) 1 + DA p| ¯ β |2 where CA = ΛA if ΛA > 0, otherwise CA = 0. Note that CA > CD , meaning that ρA is more entangled than ρD . For the case with Bob’s single-qubit quantum measurement reversal, CB is the same as CA except the fact that DA and DB are interchanged, and CA and CB give the same value when DA = DB = D. From Eq. (2), we can find several remarkable results. First of all, ΛA is monotonic increasing function of p, meaning that the resulting state ρA becomes more entangled as the weak and reversing strengths increase. Note that the concurrence of the initial state |Φi, 2 |αβ |, cannot be fully retrieved since only Alice performs single-qubit quantum measurement reversal on her qubit. However, even in the ESD condition, ρA can have non-zero entanglement if the strengths of the weak and its corresponding reversing measurements (pr = p + DA p) ¯ are larger than the threshold value, i.e., p ≥ 1 − |α /β |2 /(DA DB ). Hence, we can conclude that Alice can avoid ESD by performing single-qubit quantum measurement reversal only on her qubit without Bob’s help. Once Alice and Bob share non-zero entanglement against decoherence, they can prepare a maximally entangled state from multiple copies of less entangled states using entanglement distillation protocols [25–28]. 3.

Experiment

The experimental setup for demonstrating that Alice or Bob alone can avoid ESD and prevent complete loss of entanglement against decoherence using single-qubit quantum measurement reversal is shown in Fig. 3. Two-qubit entangled states are prepared by two-photon polarization states which are generated via spontaneous parametric down conversion process. A 6 mm type-I β -BaB2 O4 crystal is pumped by a 405 nm diode laser beam (100 mW) and a pair of down-converted photons centered at 810 nm are generated on the frequency-degenerate, noncollinear phase matching condition. Each down-converted photon is frequency-filtered using an interference filter with full width at half-maximum bandwidth of 5 nm. Then, the two-qubit entangled state |Φi = α |00i + β |11i can be prepared using quantum interferometry [31] and glass-plates oriented at Brewster angle (BPs) [20]. Here, |0i and |1i correspond to the horizontally (|Hi) and vertically (|V i) polarized photons, respectively. The weak measurement on the polarization qubit can be implemented using BPs since BP transmits horizontally polarized photons without any reflection while reflecting vertically polarized photons with some probability [20]. The reversing measurement partially collapses the state to the opposite direction, so it can be realized by adding 45◦ half-wave plates (HWPs) for bit-flip operations before and after BPs. The strengths of weak and reversing measurement p and pr can be increased by adding more BPs. The amplitude damping decoherence for the photonic system can be realized using a displaced Sagnac interferometer [7, 22, 24]. In experiment,

#214791 - $15.00 USD Received 27 Jun 2014; accepted 11 Jul 2014; published 29 Jul 2014 (C) 2014 OSA 11 August 2014 | Vol. 22, No. 16 | DOI:10.1364/OE.22.019055 | OPTICS EXPRESS 19059

Alice State preparation QST

f =300mm

405nm diode laser

IF

Type-I BBO crystal

HWP PBS

Bob

QWP BS

Mirror

BP

QST

Polarizer

Fig. 3. Experimental setup. A set of BPs and HWPs at the state preparation part is exploited for preparing the non-maximal entangled states. Weak and reversing measurements are also implemented by BPs and 45◦ HWPs. The amplitude damping channel for single qubit polarization state is realized using displaced Sagnac-type interferometer and additional beam splitters [7,22,24]. IF: interference filter, HWP: half-wave plate, QWP: quarter-wave plate, BP: Brewster-angle glass plate, PBS: polarizing beam splitter, BS: beam splitter, QST: quantum state tomography.

the system qubit is encoded in the polarization of the single photon (|Hi = |0iS , |V i = |1iS ) and the environment qubit is encoded in the path of the single photon, |0iE and |1iE , see Fig. 3. Finally, the resulting states are analyzed by quantum state tomography. 4.

Results and analysis

Before confirming whether our scheme works well, we have investigated how the initial entangled state |Φi = α |00i + β |11i is affected by the identical decoherence (DA = DB = D). We evaluated the concurrence of the final state ρD from the density matrices which are experimentally reconstructed by the quantum state tomography. For the two entangled input states with |α | = |β | and |α | = 0.481 < |β |, we measured CD (ΛD ) as increasing the strength of the decoherence D and the results are shown in Fig. 4(a). As expected, CD decreases as D increases in both cases. Note that ESD is observed for the case of |α | = 0.481. Then, consider the case that only Alice or Bob alone carries out single-qubit quantum measurement reversal to avoid complete loss of entanglement when each qubit undergoes separate decoherence. We performed the experiment for the identical damping case, i.e., DA = DB = D. In addition, to simulate the situation in which ESD can take place, we chose D = 0.617 ± 0.017. Figure 4(b) shows the experimental results for CA and CB (ΛA and ΛB ) as a function of p for two initial entangled states. We have collected experimental data up to p = 0.617 due to low success probabilities associated with weak and reversing measurements for larger p. As expected from Eq. (2), CA and CB become larger as p increases. In addition, for the input state with |α | = 0.481 which experiences ESD without introducing quantum measurement reversal, CA and CB can have non-zero values when strengths of the weak and reversing measurements are larger than the threshold value. This result demonstrates that Alice and Bob can share non-zero entanglement even in the ESD condition using only single-qubit quantum measurement reversal, i.e., ESD can be avoided by Alice or Bob alone.

#214791 - $15.00 USD Received 27 Jun 2014; accepted 11 Jul 2014; published 29 Jul 2014 (C) 2014 OSA 11 August 2014 | Vol. 22, No. 16 | DOI:10.1364/OE.22.019055 | OPTICS EXPRESS 19060

1.0 0.8 Concurrence

0.6 0.4 0.2 0 -0.2

0

0.2

0.4 0.6 Decoherence

0.8

1.0

0

0.2 0.4 0.6 0.8 Weak measurement strength

1.0

1.0

Concurrence

0.8 0.6 0.4 0.2 0 -0.2

Fig. 4. (a) Concurrence CD of the resulting state ρD decreases as the strength of the decoherence (D) increases. (b) Concurrence CA (CB ) of the final state ρA (ρB ) in the case that only Alice (Bob) perfroms single-qubit quantum measurement reversal on her (his) qubit (DA = DB = 0.617). p is the strength of the weak measurement. Solid lines are theoretical results. Negative values corresponds to ΛD for (a), and ΛA and ΛB for (b), respectively. Note that C = 0 when Λ < 0 since C = max {0, Λ}. We evaluated the fidelity between the ideally expected state and the experimentally reconstructed state for ρA and ρB . The fidelity values for ρA (FA ) and ρB (FB ) are averaged from the experimental data of (b), and we obtain FA = 0.945 ± 0.024 and FB = 0.935 ± 0.028.

5.

Conclusion

When two entangled qubits, each owned by Alice and Bob, experience separate decoherence, the amount of entanglement decreases and sometimes ESD occurs. We have demonstrated that Alice or Bob alone overcomes ESD and still share non-zero entanglement after decoherence with quantum measurement reversal on only one of the two entangled qubits. Since any small amount of entanglement shared by Alice and Bob can be concentrated, we believe that our scheme has important applications in quantum information protocols based on distributed or stored entangled qubits as they are subject to decoherence. Moreover, our scheme can be directly applied to other qubit systems experiencing decoherence, such as, two-level atoms with spontaneous decay, superconducting qubits with zero-temperature energy relaxation, and so on.

#214791 - $15.00 USD Received 27 Jun 2014; accepted 11 Jul 2014; published 29 Jul 2014 (C) 2014 OSA 11 August 2014 | Vol. 22, No. 16 | DOI:10.1364/OE.22.019055 | OPTICS EXPRESS 19061

Appendix : Reversing measurement strength under decoherence When an initial quantum state |ψin i = α |0i + β |1i does not suffer any decoherence (D = 0), √ 1 − p|1i the reversing measurement for the weak measurement W (p) = |0i h0| + h1| is √ R (pr ) = 1 − pr |0i h0|+ |1i h1| with pr = p [17]. The quantum state |ψin i = α |0i + β |1i is retrieved without errors only when pr = p, and, thus, pr = p is the optimal reversing measurement strength for D = 0. Note that the optimal pr is independent on α in this case. However, when decoherence is introduced, pr = p is not the optimal measurement anymore and the “optimality” of the reversing measurement strength becomes highly non-trivial. In general, pr is dependent on several parameters such as D, p, and α . Since the initial state cannot be retrieved perfectly, the optimal pr value is different for maximizing, e.g., the state fidelity, the success probability, the concurrence, etc. Moreover, the optimal pr value is also dependent on the condition of the decoherence suppression scenarios. In this section, we investigate several reversing measurement strategies and compare them with respect to the state fidelity and the success probability. Here, we consider three different situations i) a single qubit state suffers decoherence [21, 22], ii) both subsystems of the two-qubit entangled state suffer from decoherence (the scenarios considered in [24] and in this manuscript), and iii) one subsystem of the two-qubit entangled state suffers from decoherence. A1. Single qubit case As shown in Fig. 1(a), if a set of weak measurement W (p) and reversing measurement R(pr ) measurements are performed before and after decoherence (D), an initial qubit state |ψin i = α |0i + β |1i, where |α |2 + |β |2 = 1, is transformed to the final state ρout , given as p ! 1 p¯r |α |2 + D p| p¯r D¯ p¯αβ ∗ ¯ β |2 ρout = , (3) p PS p¯r D¯ p¯α ∗ β D¯ p| ¯ β |2

where D¯≡ 1 − D, p¯ ≡ 1− p, p¯r ≡ 1 − pr . The success probability or channel transmittance is ¯ β |2 + D¯ p| ¯ β |2 [21, 22]. PS = p¯r |α |2 + D p| One way to quantify performance of the decoherence suppression scheme is evaluating the state fidelity between the initial state ρin = |ψin i hψin | and the final state ρout , i.e., F = hψin | ρout |ψin i, which is calculated to be i h i h p 1 − p|β |4 − pr |α |4 − D p| ¯ β |2 1 + (pr − 2) |α |2 − 2|α |2 |β |2 1 − p¯r D¯ p¯ F= . (4) ¯ r |α |2 − p (1 − Dpr ) |β |2 1 − Dpr − Dp Then, the reversing measurement strength pmax value which gives the maximum fidelity value r Fmax is

pmax r

=

− 1−p|2 β | 2 D p| ¯ β | +|α | 2

+

2 2

(D p+p) ¯ (|α | −|β | 2

)

s

+4|α | −1+D¯ p¯(|α |2 −|β |2 ) 2

2|α |4

2 D p| ¯ β |2 |α |2 −|β |2 +|α |2 2 2 D p| ¯ β | +|α |

(

)

. (5)

Note that pmax depends on α , D, and p. r However, in practical decoherence scenario, we may assume that we know the magnitude of decoherence D while we do not have any prior information about the initial state |ψin i, i.e., α is an unknown parameter [22, 24]. We√ consider two different state independent pr values. First, we obtain pr by substituting |α | = 1/ 2 into Eq. (5), prfix =

2D + p − 2Dp , D + 1 − Dp

(6)

#214791 - $15.00 USD Received 27 Jun 2014; accepted 11 Jul 2014; published 29 Jul 2014 (C) 2014 OSA 11 August 2014 | Vol. 22, No. 16 | DOI:10.1364/OE.22.019055 | OPTICS EXPRESS 19062

fix where pfix r is now state-independent, meaning that pr is not a function α . exp Second, we choose pr to be pr = p + (1 − p)D which is used in our paper [21, 22]. The exp exp exp reasons we chose pr are the following: i) pr is state independent, i.e., pr is not a function of α and ii) the state fidelity F asymptotically approaches to 1 as the weak measurement strength p → 1 regardless of α . We now compare the state fidelity values F between the initial and final states for two difexp ferent reversing measurement strength pfix r and pr . The state fidelity values for the strength of exp fix the reversing measurement pr and pr , respectively, are calculated to be √ 1 + |β |2 D p¯ − 2 |αβ |2 1 − 1 + D p¯ fix , (7) F = 1 + 2D p| ¯ β |2

and F exp =

1 + D p| ¯ αβ |2 1 + D p| ¯ β |2

.

(8)

F max can be obtained by substituting pmax to Eq. (3). Note that F max is always higher than both r fix exp of F and F regardless of |α |, D, and p. However, to achieve F max , we need to know prior information about the initial state, meaning that the decoherence suppression scheme becomes state dependent. Thus, here, we focus on comparing F exp with F fix . exp The numerical results for the state fidelities for reversing strengths pfix when r and pr exp fix D = 0.617 are shown in Fig. 5(a). Note that F > F when |α | is larger than |β |. Since our decoherence suppression scheme exploits weak measurement, the success probability of the scheme is not unity. Hence, the success probability is also an important parameter for quanexp tifying the performance of the scheme. The success probabilities for pfix r and pr , respectively, (a)

1.0

(b)

1.0

0.5 0.5 0 0 1.0 0.6

0.8

0.4

0.2

0.6 1.0

0 1.0 0.5 0.5 0 0

Fig. 5. Result of the scenario in Fig. 1(a) with D = 0.617. (a) State fidelities F fix and F exp as functions of |α | and p. F fix and F exp increase as the weak measurement strength p exp exp increases. (b) Success probabilities PSfix and PS as functions of |α | and p. PS is always fix larger than PS regardless of |α | and p values. Note that F → 1 and PS → 0 as p → 1 for exp both cases of pfix r and pr .

#214791 - $15.00 USD Received 27 Jun 2014; accepted 11 Jul 2014; published 29 Jul 2014 (C) 2014 OSA 11 August 2014 | Vol. 22, No. 16 | DOI:10.1364/OE.22.019055 | OPTICS EXPRESS 19063

are given by PSfix =

D¯ p¯ 1 + 2D p| ¯ β |2 1 + D p¯

exp

and PS

= D¯ p¯ 1 + D p| ¯ β |2 .

(9)

PSmax can be obtained by substituting pmax into PS of Eq. (3). Note that, in most cases, PSmax is r exp fix max lower than PS and PS since pr is chosen for maximizing the state fidelity F. Figure 5(b) represents the success probability dependencies on |α | and p for different reversexp ing measurement strengths when D = 0.617. As shown in Fig. 5(b), PS > PSfix for all values of |α | and p. In the single qubit case, the reversing measurement strength pexp r = p + (1 − p)D is more efficient than pmax and pfix r r in terms of the success probability. A2. Two-qubit case: Both subsystems suffer from decoherence and two pairs of weak and reversing measurements are performed We consider the situation shown in Fig. 1(b) in which the initial two-qubit entangled state |Φi = α |00i + β |11i, where |α |2 + |β |2 = 1, suffers from separate decoherence and two pairs of weak and reversing measurements are applied to suppress decoherence [24]. To simplify the situation, we assume that each subsystem undergoes the same magnitude of decoherence (DA = DB = D) and the strengths of weak measurements WA (p) and WB (p), and reversing measurements RA (pr ) and RB (pr ) performed on each subsystem are the same. Then, the output state ρout is given as 2 ¯ r αβ ∗ |α | + D2 |β |2 p¯2 p2r 0 0 p¯Dp 1 0 0 D |β |2 D¯ p¯2 pr 0 ρout = , (10) 2 ¯ 2 PS 0 0 0 D |β | D p¯ pr 2 ∗ 2 2 ¯ rα β p¯Dp 0 0 |β | D¯ p¯ where PS = pr D¯ |α |2 (Dpr + pr − 2) + (Dpr − 1)2 1 − p2 |β |2 + 2p |β |2 is the success probability. The state fidelity between the initial state and the output state is obtained to be h i h i h i2 |β |2 D2 p¯2 |α |2 (pr − 2) pr + 1 − 2 |β |2 D p¯ p¯ + |α |2 (p − pr ) + p¯ + |α |2 (p − pr ) h i , (11) F= ¯ r [(D + 1)pr − 2] + |β |2 (p − 2)p + 1 (Dpr − 1) 2 |α |2 Dp and the reversing measurement strength maximizing F is r |α |2 −D p| ¯ β |2 (4|α |2 −D p¯) ¯ D p ¯ 2 ¯ |α |2 +D2 p¯ 2 |β |2 D p¯ + 2p |α | pmax = − , r 2 2 |α | (1 − D p) 2 |α |2 (1 − D p) ¯ ¯

(12)

if i) |α | ≥ √12 or ii) |α | < √12 and D p¯ |β |2 < |α |2 < p¯ |β |2 . Otherwise, pr = 1 maximizes F, meaning that the reversing measurement pr should be projection measurement regardless of the weak measurement strength p. √ As we did in the single qubit case, pfix r is obtained by substituting α = 1/ 2 into Eq. (12) and we obtain p D2 p¯2 + 1 − D¯ p¯ p , (13) pfix = r D2 p¯2 + 1 #214791 - $15.00 USD Received 27 Jun 2014; accepted 11 Jul 2014; published 29 Jul 2014 (C) 2014 OSA 11 August 2014 | Vol. 22, No. 16 | DOI:10.1364/OE.22.019055 | OPTICS EXPRESS 19064

(a)

(b)

1.0 0.4 0.8 1.0

1.0

0.6 0.4

0.5

0.2

0.5 0 1.0

1.0 0.5

0.5 0

0

Fig. 6. Result of the scenario in Fig. 1(b) with DA = DB = 0.617. (a) State fidelities F fix and F exp as functions of |α | and p. The state fidelities increase as the weak measurement exp strength p increases. (b) Success probabilities PSfix and PS as functions of |α | and p. Note that PS → 0 as p → 1 for all both cases.

and the corresponding state fidelity F fix for pfix r is given as p D2 p¯2 + 1 − 1 + D2 p¯2 |β |2 + 1 2 | α |2 |β |2 p . F fix = 2D p¯ |β |2 D2 p¯2 + 1 + D p¯ + 1 exp

On the other hand, the state fidelity F exp for pr F exp =

(14)

= p + (1 − p)D is calculated to be

1 + D2 p¯2 |α |2 |β |2

1 + D p| ¯ β |2 (D p¯ + 2)

.

(15) exp

The numerical results for the state fidelities F for pfix are shown in Fig. 6(a). As r and pr shown in Fig. 6(a), the general tendency of F fix and F exp are similar to the results of the single qubit case, see Fig. 5(a). F exp is generally larger than F fix when |α | is larger than |β |. exp Finally, the success probabilities for pfix r and pr , respectively, are calculated to be ! 2 2 p¯ 2 |β |2 + |α |2 2D p ¯ | D β | 2 fix 2 2 +p + |β | , PS = D¯ p¯ (16) D2 p¯2 + 1 D2 p¯2 + 1

and

exp

PS

h i = D¯ 2 p¯2 1 + D p| ¯ β |2 (D p¯ + 2) .

(17)

Figure 6(b) shows the success probabilities PS for various reversing measurement strength exp pr . As shown in Fig. 6(b), PS is always larger than PSfix in any combination of |α | and p. The exp general tendency of the success probabilities are similar to the single qubit case in that PS is fix larger than PS . A3. Two-qubit case: Both subsystems suffer from decoherence and one pair of weak and reversing measurements are performed In this section, we consider the scenario of this paper, see Fig. 1(c). Each subsystem of the initial entangled state |Φi = α |00i + β |11i, where |α |2 + |β |2 = 1, undergoes separate decoherence #214791 - $15.00 USD Received 27 Jun 2014; accepted 11 Jul 2014; published 29 Jul 2014 (C) 2014 OSA 11 August 2014 | Vol. 22, No. 16 | DOI:10.1364/OE.22.019055 | OPTICS EXPRESS 19065

and a set of weak measurement WA (p) and reversing measurement RA (pr ) is performed only on the subsystem undergoing decoherence (A). If we assume each subsystem undergoes the identical decoherence (DA = DB = D), the output state ρout is given as √ √ p¯r D2 p¯ |β |2 + |α |2 0 0 αβ ∗ D¯ p¯ p¯r 1 0 DD¯ p¯ |β |2 p¯r 0 0 (18) ρout = , 2 PS ¯ 0 0 DD p¯ |β | 0 √ √ 0 α ∗ β D¯ p¯ p¯r 0 D¯ 2 p¯ |β |2

where PS = |α |2 [p − pr (D¯ + Dp)] + p¯ (1 − Dpr ) and the corresponding state fidelity between the initial state and the final state is calculated to be

F=

√ √ D2 p| ¯ β |2 (1−|α |2 pr )−2|α |2 ( p¯ p¯r −1)(D|β |2 −1)−|α |4 (2 p¯ p¯ r +pr −2)−2D|β |2 +(2D−1)p|β |4 +1 . ¯ α |2 pr ( p|β |2 −1)(Dpr −1)−D|

(19)

Since the reversing measurement strength pmax which maximizes F is rather complicated and r max . pfix is calculated to be cannot be simplified, we provide pfix instead of p r r r r h i (2D p¯ + p) 2 − D p¯ D (2D p¯ + p) + D D (2D p¯ + p)2 − 4 p¯ − 4p + 4 − 2 . (20) pfix r = 2 (D p¯ + 1)2 Since the final expressions of F fix and PSfix are also complicated, we omit their analytic exexp exp pressions. F exp and PS values for pr , respectively, are given as √ D¯ − 1 |α |2 |β |2 − D |β |4 + 1 D2 p¯ |α |2 |β |2 + 2 F exp = , (21) D p¯ |β |2 + 1 and exp

PS (a)

= D¯ p¯ D p¯ |β |2 + 1 .

(22)

(b)

1.0 0.5

0 1.0

0.8

0.5 0.4

1.0 0 1.0 0.5 0.5

0 1.0 0.5

0 0

Fig. 7. Result of the scenario in Fig. 1(c) with DA = DB = 0.617. (a) State fidelities F fix and F exp as functions of |α | and p. The state fidelities increase as the weak measurement exp strength p increases. (b) Success probabilities PSfix and PS as functions of |α | and p. PSfix exp is always larger than PS regardless of |α | and p values. Note that F does not approach to exp unity and PS → 0 as p → 1 for both cases of pfix r and pr . #214791 - $15.00 USD Received 27 Jun 2014; accepted 11 Jul 2014; published 29 Jul 2014 (C) 2014 OSA 11 August 2014 | Vol. 22, No. 16 | DOI:10.1364/OE.22.019055 | OPTICS EXPRESS 19066

Although we do not provide explicit forms of F fix and PSfix due to their complexity, we show exp numerical results to compare F fix and PSfix with F exp and PS , respectively. Figure 7(a) shows the state fidelity results for various reversing strengths. Since a single pair of weak and reversing measurements cannot suppress two independent decoherence, the state fidelities F cannot achieve unity even in the case that p → 1. Note that F → 1 as p → 1 in both of Fig. 5(a) and Fig. 6(b). Another thing to note is that F fix is larger than F exp when |α | is larger than |β | in Fig. 7(a) while F exp is larger than F fix when |α | is larger than |β | in both of Fig. 5(a) and Fig. 6(a). Interestingly, the difference between F fix , and F exp is very small in any combination of |α | and p in Fig. 7(a). The general tendency of the success probabilities in this case are opposite to the previous exp exp cases, see Fig. 7(b). PSfix is always larger than PS in Fig. 7(b); however, PS is always larger fix than PS in both of Fig. 5(b) and Fig. 6(b). A4. Two-qubit case: One subsystem suffers from decoherence and one pair of weak and reversing measurements are performed Finally, we consider the case of Fig. 1(c) with DA = D and DB = 0. Here only the subsystem A of the initial entangled state |Φi = α |00i+ β |11i, where |α |2 + |β |2 = 1, suffers from decoherence and a set of weak measurement WA (p) and reversing measurement RA (pr ) is performed on the subsystem A. Then, the output state ρout becomes √ √ √ |α |2 p¯r D¯ p¯ p¯r αβ ∗ 0 0 1 0 D |β |2 p¯ p¯r 0 0 , ρout = (23) 0 0 0 0 PS √ √ √ D¯ p¯ p¯r α ∗ β 0 0 |β |2 D¯ p¯

where PS = |β |2 D¯ p¯ + |β |2 D p¯ p¯r + |α |2 p¯r and the state fidelity between the input state and the output state is given as p p D¯ p¯ p¯r − 1 + p |β |4 − 1 D p¯ |β |4 + |α |4 2 D¯ p¯ p¯r + pr − 2 − 2 |α |2 , (24) F= D¯ |α |2 pr + p |β |2 − 1 (1 − Dpr ) and the reversing measurement strength pmax that maximizes F is D2 p¯2 |β |4 − D p¯ |α |2 |β |2 + 1 + p |α |4 . pmax = 2 r 2 2 |α | − D p¯ |β | √ fix Then, pfix r is evaluated by substituting α = 1/ 2 into Eq. (25) and pr is given as pfix r =

D2 p¯2 − 3D p¯ + p (D p¯ − 1)2

.

(25)

(26)

exp

The state fidelity values for pfix r and pr , respectively, are given as 2 D p¯ |β |2 + 1 1 F fix = , and F exp = . 2 D p¯ |β | (D p¯ + 3) + 1 D p¯ |β |2 + 1

(27)

exp

The success probabilities for pfix r and pr , respectively, are given as h i D¯ p¯ D p¯ |β |2 (D p¯ + 3) + 1 exp ¯ p¯ D p¯ |β |2 + 1 . , and P = D PSfix = S (D p¯ + 1)2

(28)

#214791 - $15.00 USD Received 27 Jun 2014; accepted 11 Jul 2014; published 29 Jul 2014 (C) 2014 OSA 11 August 2014 | Vol. 22, No. 16 | DOI:10.1364/OE.22.019055 | OPTICS EXPRESS 19067

(b)

(a)

1.0

0.5 1.0 0 0.6 0.8 0.4 0.6

1.0

0.2

1.0 0 1.0

0.5 0.5 0.5 0 0

Fig. 8. Result of the scenario in Fig. 1(c) with DA = 0.617 and DB = 0. (a) State fidelities exp F fix and F exp as functions of |α | and p. (b) Success probabilities PSfix and PS as functions exp of |α | and p. PS is always larger than PSfix regardless of |α | and p values. Note that F → 1 exp and PS → 0 as p → 1 for both cases of pfix r and pr .

The numerical results for the state fidelities and the success probabilities for various reversing measurement strengths are shown in Fig. 8. In this scheme, the general tendencies of the state fidelities and the success probabilities are similar to the single qubit case shown in Fig. 5, i.e., exp F exp > F fix when |α | is larger than |β | and PS > PSfix regardless of |α | and p. A5. Conclusion In conclusion, we have investigated the state fidelities F and the success probabilities PS for various decoherence suppression schemes. Our results suggest that there is no reversing measurement which maximizes the state fidelity and the success probability simultaneously. In many cases, a trade-off relation between F and PS holds, in other words, the higher the state fidelity, the lower the success probability, vice versa. Therefore, one should choose a proper pr value for a specific application, after carefully considering the trade-off relation. Acknowledgments This work was supported in part by the National Research Foundation of Korea (Grant No. 2013R1A2A1A01006029 and 2011-0021452). H.-T.L. and J.-C.L. acknowledge support from the National Junior Research Fellowship (Grant No. 2012-000642 and 2012-000741, respectively).

#214791 - $15.00 USD Received 27 Jun 2014; accepted 11 Jul 2014; published 29 Jul 2014 (C) 2014 OSA 11 August 2014 | Vol. 22, No. 16 | DOI:10.1364/OE.22.019055 | OPTICS EXPRESS 19068